Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94694 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7418 |
| commutator | C2×C32 | C2×C4 |
| Frattini, cosocle | C2×M64 | C22 |
| prime radical | #13697 | C2 |
| LCS | G ▷ C2×C32 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ C22×C16 ◁ C2×M64 ◁ G | |
| LECS | G ▷ C2×M64 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
91 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,4,12,9,15,8,32)(3,10,18,33,21,37,22,36)(5,26,23,57,6,29,24,58)(7,30,25,48,31,45,28,14)(11,38,39,19,42,54,41,16)(13,44,43,27,46,53,60,17)(20,47,49,35,56,63,50,61)(34,64,62,51,59,55,40,52);(2,10)(3,16)(4,8)(5,27)(6,17)(7,28)(11,13)(12,33)(14,47)(15,37)(18,54)(19,21)(20,55)(22,38)(23,44)(24,53)(25,31)(26,59)(29,34)(30,35)(32,36)(39,43)(40,57)(41,60)(42,46)(45,61)(48,63)(49,51)(50,52)(56,64)(58,62)Polycyclic presentation:
2^9|a^2=d,c^2=fgi,d^2=i,e^2=ghi,f^2=ghi,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fgi,g^a=gh,h^a=hi,c^b=cf,d^b=defgi,e^b=egi,f^b=fgi,g^b=gh,h^b=hi,d^c=di53 conjugacy classes — view the full character table.
| maximal subgroup of | #72225468 |
| maximal normal subgroup of | #72225468 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1892);